In mathematics, a polycyclic group is a solvable group

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

 that satisfies the maximal condition on subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s (that is, every subgroup is finitely generated

Finitely generated

In mathematics, finitely generated may refer to:* Finitely generated group* Finitely generated monoid* Finitely generated abelian group* Finitely generated module* Finitely generated ideal* Finitely generated algebra* Finitely generated space...

). Polycyclic groups are finitely presented, and this makes them interesting from a computational point of view.

Terminology

Equivalently, a group G is polycyclic if and only if it admits a subnormal series with cyclic factors, that is a finite set of subgroups, let's say G₀, ..., G_n such that

• G₀ coincides with G
• G_n is the trivial subgroup
• G_i+1 is a normal subgroup of G_i (for every i between 0 and n - 1)
• and the quotient group G_i / G_i+1 is a cyclic group
  Cyclic group
  In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
  
   (for every i between 0 and n - 1)

A metacyclic group

Metacyclic group

In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. That is, it is a group G for which there is a short exact sequence1 \rightarrow K \rightarrow G \rightarrow H \rightarrow 1,\,where H and K are cyclic...

 is a polycyclic group with n ≤ 2, or in other words an extension

Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...

 of a cyclic group by a cyclic group.

Examples

Examples of polycyclic groups include finitely generated abelian groups, finitely generated nilpotent

Nilpotent group

In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...

 groups, and finite solvable groups. Anatoly Maltsev

Anatoly Maltsev

Anatoly Ivanovich Maltsev was born in Misheronsky, near Moscow, and died in Novosibirsk, USSR. He was a mathematician noted for his work on the decidability of various algebraic groups...

 proved that solvable subgroups of the integer general linear group

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

 are polycyclic; and later Louis Auslander

Louis Auslander

Louis Auslander was an American mathematician. He had wide ranging interests both in pure and applied mathematics and worked on Finsler geometry, geometry of solvmanifolds and nilmanifolds, locally affine spaces, many aspects of harmonic analysis, representation theory of solvable Lie groups, and...

 (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices. The holomorph

Holomorph (mathematics)

In mathematics, especially in the area of algebra known as group theory, the holomorph of a group is a group which simultaneously contains the group and its automorphism group. The holomorph provides interesting examples of groups, and allows one to treat group elements and group automorphisms in...

 of a polycyclic group is also such a group of integer matrices.

Strongly polycyclic groups

A group G is said to be strongly polycyclic if it is polycyclic with the added stipulation that each G_i / G_i+1 is infinitely cyclic. Clearly, a strongly polycyclic group is polycyclic. Also, any subgroup of a strongly polycyclic group is strongly polycyclic.

Polycyclic-by-finite groups

A virtually polycyclic group is a group that has a polycyclic subgroup of finite index, an example of a virtual

Virtually

In mathematics, especially in the area of abstract algebra which studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index...

 property. Such a group necessarily has a normal polycyclic subgroup of finite index, and therefore such groups are also called polycyclic-by-finite groups. Although polycyclic-by-finite groups need not be solvable, they still have many of the finiteness properties of polycyclic groups; for example, they satisfy the maximal condition, and they are finitely presented and residually finite

Residually finite group

In the mathematical field of group theory, a group G is residually finite or finitely approximable if for every nontrivial element g in G there is a homomorphism h from G to a finite group, such thath \neq 1.\,...

.

In the textbook and some papers, an M-group refers to what is now called a polycyclic-by

Group extension

In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence...

-finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

, which by Hirsch's theorem can also be expressed as a group which has a finite length subnormal series with each factor a finite group or an infinite cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

.

These groups are particularly interesting because they are the only known examples of Noetherian

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

 group ring

Group ring

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...

s , or group rings of finite injective dimension.

Hirsch length

The Hirsch length or Hirsch number of a polycyclic group G is the number of infinite factors in its subnormal series.

If G is a polycyclic-by-finite group, then the Hirsch length of G is the Hirsch length of a polycyclic normal subgroup

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

 H of G, where H has finite index in G. This is independent of choice of subgroup, as all such subgroups will have the same Hirsch length.